Kristallchemie und Kristallstrukturdatenbanken Pulverdiffraktometrie Einkristall Strukturanalyse Strukturanalyse mittels Pulverdaten Kristallchemie in der Strukturanalyse Modellbau Simulated annealing Evolutionäre Algorithmen FOCUS Charge flipping

Simulated annealing und Zeolithe M.W. Deem and J.M. Newsam, "Determination of 4-connected framework crystal structures by simulated annealing" Nature 342, 260-262 (1989) M. Falcioni and M.W. Deem, "A biased Monte Carlo scheme for zeolite structure solution" J. Chem. Phys. 110, 1754-1766 (1999)

Simulated annealing und Zeolithe

Simulated annealing und Zeolithe T-T Abstände T T-T-T Winkel Anzahl nächste Nachbaren Pulverdiagramm  figure of merit (χ2) zufällige Verschiebung aller Atome

Simulated annealing und Zeolithe "Move" akzeptiert wenn T χ2neu < χ2alt oder n < e-δ δ = (χ2neu-χ2alt )/T χ2neu-χ2alt klein und/oder T gross  δ klein  e-δ gross  "Move" eher akzeptiert

Simulated annealing und Zeolithe

Simulated annealing und Molekülstrukturen chemische Zusammensetzung C10H16N6S Verknüpfung Bindungslängen, Bindungswinkel, Torsionswinkel Cimetidine

Simulated annealing und Molekülstrukturen HN HN S HN N N Cimetidine N Molekül kann mittels interne Koordinaten beschrieben werden C - C 1.36 Å C - C 1.49 Å C - C - C 120˚ C - S 1.82 Å C - C - S 109.5˚ C - S 1.82 Å C - C - S 109.5˚ C - C - C - S 180˚

Simulated annealing und Molekülstrukturen HN HN S HN N N Cimetidine N Molekül kann mittels interne Koordinaten beschrieben werden Parameter Position des Moleküls X,Y,Z Orientierung des Moleküls Θ, Φ, Ψ freie Torsionswinkel τ1, τ2, τ3, τ4, τ5, τ6, τ7 Total: 13 statt 17 x 3 = 51 Atomkoordinaten

Simulated annealing und Molekülstrukturen (1) Start mit einem Satz Strukturparameter φalt {X,Y,Z,Θ,Φ,Ψ,τ1-n} Struktur chemisch sinnvoll (2) Figure-of-merit (z.B. R-Wert) rechnen χ2 alt Kann auch andere Kriterien berücksichtigen z.B. Coulomb Potentiale (3) Strukturparameter modifizieren φneu = φalt + m*Δφalt m ist ein Zufallszahl zwischen 0 und 1 (4) Neuer Figure-of-merit rechnen χ2neu (5) Wenn χ2neu < χ2alt oder φneu  φalt n < exp (-(χ2neu - χ2alt) / T) sonst φalt unverändert n ist ein Zufallszahl zwischen 0 und 1 ermöglicht Herauskommen aus falsche Minima (6) Nachdem die vorgeschriebene Anzahl T reduziert "Moves" akzeptiert Annealing Schema weniger Strukturen mit χ2neu > χ2alt akzeptiert (7) Zurück zu Schritt (3)  optimierte Struktur

Simulated Annealing Initial model SA control parameters http://vincefn.net/Fox/FoxWiki Random variation of X, Y, Z, Θ, Φ, Ψ, τn Is the model chemically reasonable? no Back to previous model Reduce temperature yes yes Powder data Evaluate fitness Move acceptable? yes Prescribed number of moves reached? no no

Polymer Clarifier (C) H C7H3 || | | || | | (C) = C1 - C2 – C3 – N5 – C6 – C8H3 || | O4 C9H3 1.39 Å 1.50 Å 1.45 Å 1.33 Å 1.53 Å 1.23 Å 1 2 3 5 6 7 8 9 4 No. Atom Distance Angle τ A shiftmax τmin τmax B 1 C1 2 C2 3 C3 4 O4 5 N5 6 C6 7 C7 8 C8 9 C9

Polymer Clarifier (C) H C7H3 || | | || | | (C) = C1 - C2 – C3 – N5 – C6 – C8H3 || | O4 C9H3 1.39 Å 1.50 Å 1.45 Å 1.33 Å 1.53 Å 1.23 Å 1 2 3 5 6 7 8 9 4 No Atom Distance Angle τ A shiftmax τmin τmax B 1 C1 2 C2 1.39 Å 3 C3 4 O4 5 N5 6 C6 7 C7 8 C8 9 C9

Polymer Clarifier (C) H C7H3 || | | || | | (C) = C1 - C2 – C3 – N5 – C6 – C8H3 || | O4 C9H3 1.39 Å 1.50 Å 1.45 Å 1.33 Å 1.53 Å 1.23 Å 1 2 3 5 6 7 8 9 4 No Atom Distance Angle τ A shiftmax τmin τmax B 1 C1 2 C2 1.39 Å 3 C3 1.50 Å 120˚ 4 O4 5 N5 6 C6 7 C7 8 C8 9 C9

Polymer Clarifier (C) H C7H3 || | | || | | (C) = C1 - C2 – C3 – N5 – C6 – C8H3 || | O4 C9H3 1.39 Å 1.50 Å 1.45 Å 1.33 Å 1.53 Å 1.23 Å 1 2 3 5 6 7 8 9 4 No Atom Distance Angle τ A shiftmax τmin τmax B 1 C1 2 C2 1.39 Å 3 C3 1.50 Å 120˚ 4 O4 1.23 Å 0˚ 5 N5 6 C6 7 C7 8 C8 9 C9

Polymer Clarifier (C) H C7H3 || | | || | | (C) = C1 - C2 – C3 – N5 – C6 – C8H3 || | O4 C9H3 1.39 Å 1.50 Å 1.45 Å 1.33 Å 1.53 Å 1.23 Å 1 2 3 5 6 7 8 9 4 No Atom Distance Angle τ A shiftmax τmin τmax B 1 C1 2 C2 1.39 Å 3 C3 1.50 Å 120˚ 4 O4 1.23 Å 0˚ 5 N5 1.33 Å 180˚ 6 C6 7 C7 8 C8 9 C9

Polymer Clarifier (C) H C7H3 || | | || | | (C) = C1 - C2 – C3 – N5 – C6 – C8H3 || | O4 C9H3 1.39 Å 1.50 Å 1.45 Å 1.33 Å 1.53 Å 1.23 Å 1 2 3 5 6 7 8 9 4 No Atom Distance Angle τ A shiftmax τmin τmax B 1 C1 2 C2 1.39 Å 3 C3 1.50 Å 120˚ 4 O4 1.23 Å 0˚ 5 N5 1.33 Å 180˚ 6 C6 1.45 Å 7 C7 8 C8 9 C9

Polymer Clarifier (C) H C7H3 || | | || | | (C) = C1 - C2 – C3 – N5 – C6 – C8H3 || | O4 C9H3 1.39 Å 1.50 Å 1.45 Å 1.33 Å 1.53 Å 1.23 Å 1 2 3 5 6 7 8 9 4 No Atom Distance Angle τ A shiftmax τmin τmax B 1 C1 2 C2 1.39 Å 3 C3 1.50 Å 120˚ 4 O4 1.23 Å 0˚ 5 N5 1.33 Å 180˚ 6 C6 1.45 Å 7 C7 1.53 Å 109.5˚ 60˚ 8 C8 9 C9 300˚

Polymer Clarifier (C) H C7H3 || | | || | | (C) = C1 - C2 – C3 – N5 – C6 – C8H3 || | O4 C9H3 1.39 Å 1.50 Å 1.45 Å 1.33 Å 1.53 Å 1.23 Å 1 2 3 5 6 7 8 9 4 No Atom Distance Angle τ A shiftmax τmin τmax B 1 C1 2 C2 1.39 Å 3 C3 1.50 Å 120˚ 4 O4 1.23 Å 0˚ 5 N5 1.33 Å 180˚ 6 C6 1.45 Å 7 C7 1.53 Å 109.5˚ 60˚ 8 C8 9 C9 300˚

Polymer Clarifier (C) H C7H3 || | | || | | (C) = C1 - C2 – C3 – N5 – C6 – C8H3 || | O4 C9H3 1.39 Å 1.50 Å 1.45 Å 1.33 Å 1.53 Å 1.23 Å 1 2 3 5 6 7 8 9 4 No Atom Distance Angle τ A shiftmax τmin τmax B 1 C1 2 C2 1.39 Å 3 C3 1.50 Å 120˚ 4 O4 1.23 Å 0˚ 5 N5 1.33 Å 180˚ 6 C6 1.45 Å 7 C7 1.53 Å 109.5˚ 60˚ 8 C8 9 C9 300˚

Polymer Clarifier (C) H C7H3 || | | || | | (C) = C1 - C2 – C3 – N5 – C6 – C8H3 || | O4 C9H3 1.39 Å 1.50 Å 1.45 Å 1.33 Å 1.53 Å 1.23 Å 1 2 3 5 6 7 8 9 4 No Atom Distance Angle τ A shiftmax τmin τmax B 1 C1 2 C2 1.39 Å 3 C3 1.50 Å 120˚ 4 O4 1.23 Å 0˚ 5 N5 1.33 Å 180˚ 6 C6 1.45 Å 7 C7 1.53 Å 109.5˚ 60˚ 8 C8 9 C9 300˚

Polymer Clarifier (C) H C7H3 || | | || | | (C) = C1 - C2 – C3 – N5 – C6 – C8H3 || | O4 C9H3 1.39 Å 1.50 Å 1.45 Å 1.33 Å 1.53 Å 1.23 Å 1 2 3 5 6 7 8 9 4 No Atom Distance Angle τ A shiftmax τmin τmax B 1 C1 2 C2 1.39 Å 3 C3 1.50 Å 120˚ 4 O4 1.23 Å 0˚ 5 N5 1.33 Å 180˚ 6 C6 1.45 Å 7 C7 1.53 Å 109.5˚ 60˚ 8 C8 9 C9 300˚

Polymer Clarifier (C) H C7H3 || | | || | | (C) = C1 - C2 – C3 – N5 – C6 – C8H3 || | O4 C9H3 1.39 Å 1.50 Å 1.45 Å 1.33 Å 1.53 Å 1.23 Å 1 2 3 5 6 7 8 9 4 No Atom Distance Angle τ A shiftmax τmin τmax B 1 C1 2 C2 1.39 Å 3 C3 1.50 Å 120˚ 4 O4 1.23 Å 0˚ 5˚ 5 N5 1.33 Å 180˚ 6 C6 1.45 Å 1˚ 7 C7 1.53 Å 109.5˚ 60˚ 8 C8 9 C9 300˚

Polymer Clarifier (C) H C7H3 || | | || | | (C) = C1 - C2 – C3 – N5 – C6 – C8H3 || | O4 C9H3 1.39 Å 1.50 Å 1.45 Å 1.33 Å 1.53 Å 1.23 Å 1 2 3 5 6 7 8 9 4 No Atom Distance Angle τ A shiftmax τmin τmax B 1 C1 2 C2 1.39 Å 3 C3 1.50 Å 120˚ 4 O4 1.23 Å 0˚ 5˚ 360˚ 5 N5 1.33 Å 180˚ -180˚ 6 C6 1.45 Å 1˚ 160˚ 200˚ 7 C7 1.53 Å 109.5˚ 60˚ 30˚ 150˚ 8 C8 270˚ 9 C9 300˚ -90˚

1 2 3 4 5 6 7 9 10 11 12 13 14 15 17 16 8 No Atom Distance Angle τ A 1.36 Å 1.35 Å 1.51 Å 1.31 Å 1.34 Å 1.52 Å 1.81 Å 1.53 Å 1.47 Å 1.37 Å 1.32 Å 1.14 Å No Atom Distance Angle τ A shiftmax τmin τmax B 1 C1 2 N2 3 C3 4 N4 5 C5 6 N6 7 N7 8 C8 9 C9 10 S10

1 2 3 4 5 6 7 9 10 11 12 13 14 15 17 16 8 No Atom Distance Angle τ A 1.36 Å 1.35 Å 1.51 Å 1.31 Å 1.34 Å 1.52 Å 1.81 Å 1.53 Å 1.47 Å 1.37 Å 1.32 Å 1.14 Å No Atom Distance Angle τ A shiftmax τmin τmax B 1 C1 2 N2 1.47 3 C3 1.37 4 N4 1.32 5 C5 1.36 6 N6 1.14 7 N7 8 C8 9 C9 1.53 10 S10 1.81

1 2 3 4 5 6 7 9 10 11 12 13 14 15 17 16 8 No Atom Distance Angle τ A 1.36 Å 1.35 Å 1.51 Å 1.31 Å 1.34 Å 1.52 Å 1.81 Å 1.53 Å 1.47 Å 1.37 Å 1.32 Å 1.14 Å No Atom Distance Angle τ A shiftmax τmin τmax B 1 C1 2 N2 1.47 3 C3 1.37 120 4 N4 1.32 5 C5 1.36 6 N6 1.14 180 7 N7 8 C8 9 C9 1.53 109.5 10 S10 1.81

1 2 3 4 5 6 7 9 10 11 12 13 14 15 17 16 8 No Atom Distance Angle τ A 1.36 Å 1.35 Å 1.51 Å 1.31 Å 1.34 Å 1.52 Å 1.81 Å 1.53 Å 1.47 Å 1.37 Å 1.32 Å 1.14 Å No Atom Distance Angle τ A shiftmax τmin τmax B 1 C1 2 N2 1.47 3 C3 1.37 120 4 N4 1.32 5 C5 1.36 6 N6 1.14 180 7 N7 8 C8 9 C9 1.53 109.5 10 S10 1.81

1 2 3 4 5 6 7 9 10 11 12 13 14 15 17 16 8 No Atom Distance Angle τ A 1.36 Å 1.35 Å 1.51 Å 1.31 Å 1.34 Å 1.52 Å 1.81 Å 1.53 Å 1.47 Å 1.37 Å 1.32 Å 1.14 Å No Atom Distance Angle τ A shiftmax τmin τmax B 1 C1 2 N2 1.47 3 C3 1.37 120 4 N4 1.32 1˚ 360 5 C5 1.36 6 N6 1.14 180 7 N7 540 8 C8 9 C9 1.53 109.5 5˚ 10 S10 1.81

1 2 3 4 5 6 7 9 10 11 12 13 14 15 17 16 8 No Atom Distance Angle τ A 1.36 Å 1.35 Å 1.51 Å 1.31 Å 1.34 Å 1.52 Å 1.81 Å 1.53 Å 1.47 Å 1.37 Å 1.32 Å 1.14 Å No Atom Distance Angle τ A shiftmax τmin τmax B 11 C11 1.81 100 180 6 5˚ 180˚ 540˚ 10 12 C12 1.51 109.5 7 13 N13 1.34 126 8 0˚ 360˚ 14 C14 1.31 108 15 N15 1.35 16 C16 17 C17 1.52

Evolutionäre Algorithmen alternativer "global optimization" Verfahren Strukturparameter sind die Gene X,Y,Z,Θ,Φ,Ψ,τ1-n Satz von Strukturparameter ist ein Chromosom {X,Y,Z,Θ,Φ,Ψ,τ1-n} Start mit einem Anzahl verschieden Individuen Neue Generation erzeugt via Rekombination/Mutation Nur die "fittest" überleben Neue Generation erzeugt ...

Evolutionäre Algorithmen - Prinzipien Parameterisierung Algorithmus zur Erzeugung eines "Phenotyps" Erzeugung einer Population möglicher Lösungen Rekombination/ Mutation Berechnung der individuellen Fitness "Survival of the fittest" R = 0.25 R = 0.31 R = 0.22 R = 0.22 R = 0.35 R = 0.42 R = 0.3

Strukturlösung mittels Modelbau (Zufälliges) Modell vom Computer Modell optimieren Methode der Optimierung least-squares refinement simulated annealing evolutionary algorithm lokal Optimierung } global Optimierung

Strukturlösung mittels Modelbau Least squares refinement Least squares is like dropping a kangaroo somewhere on the surface of the earth, telling it to hop only uphill and hoping it will get to the top of mount Everest

Strukturlösung mittels Modelbau Simulated annealing Simulated Annealing is like doing the same, but getting the kangaroo very, very drunk first.

Strukturlösung mittels Modelbau Genetic algorithms Genetic Algorithms are like taking a whole plane load of kangaroos and letting them reproduce freely (not pictured)...

Strukturlösung mittels Modelbau Genetic algorithms Note: no kangaroos were harmed in the making of this presentation and regularly shooting those at lower altitudes.